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Question #41407

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{bmatrix} 1 & 1 & 1 \\ 1 & x & x^2 \\ 2x & 3x^2 & 4x^3 \end{bmatrix}$ . Let $I_1 = \int_0^1 \det(A(x)) \, dx$ and $I_2 = \det\left( \int_0^1 A(x) \, dx \right)$ . Find the value of $900(I_1 + I_2)$ .

Options

A.
30
B.
0
C.
5
D.
-25

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Linearity and Partitioning|Linear Dependency of Rows|Elementary Row Operation Invariance

Why This Question Is Tricky

Common Mistakes

This question targets the lethal collision between elementary row operations and element-wise matrix integration.

A common trap is assuming the linear dependency of the integral operator allows $\int \det(A(x)) dx = \det(\int A(x) dx)$ , leading students to incorrectly conclude $I_1 = I_2 = 0$ .

Another severe trap occurs if students apply $x$ -dependent row operations (e.g., $R_3 \to R_3 - 2x R_2$ ) to simplify $A(x)$ before calculating $I_2$ .

While such operations preserve the determinant's value point-wise, they alter the 'weight' of the matrix elements, yielding an incorrect integrated matrix with a non-zero determinant ( $-1/36$ ), testing the invariance limits of row operations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Linearity and Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free